220 Sir Wittiam Rowan Hamuvton’s Researches respecting Quaternions. 
In fact, if we examine the changes of ordinal relation which have been made, 
in passing from the form (106) to the form (107), we shall perceive that they 
may be said to consist in first inverting the second constituent relation of the 
couple, namely, a; — a, which thus becomes a, — Aj, and in then transposing 
the two constituent relations. But this is precisely the process of ordinal deri- 
vation which was indicated in the sixth article by the characteristic R_, ,, and 
which we saw to be a symbolic square root of — 1. Indeed, as was noticed in 
that sixth article, it was on this property of this mode of derivation, that the 
present writer proposed, in a former Essay, to found a theory of algebraic couples, 
and of the use of the symbol »/(—1) in algebra. 
12. Proceeding on a similar plan, though not precisely by the formula (104), 
to illustrate those new symbolic fourth roots of unity which enter into the pre- 
sent theory of algebraic quaternions, by regarding those roots as certain charac- 
teristics of ordinal derivation, which are connected with certain other characteristics 
of momental transposition, we are now to consider a momental octad, which we 
shall denote as follows : 
C9) SS ((A Mg AG Eon, INS EG 5 Ae) 2 (108) 
and shall regard as being connected, on the plan of the tenth article, with the 
ordinal quaternion, 
q = (Aos Aly Ads A) - (Av Ay Ass Ay) 3 (109) 
that is, by (48) and (49), with the ordinal quaternion (55). If we operate on 
the octad Q by the characteristic of transposition w, defined by the symbolic 
equation (20) of the second article, then, according to a remark lately made, the 
resulting octad wQ corresponds to, or is (on the present plan) connected with, the 
quaternion — q; and thus the two signs » and —, as here used, have a certain 
correspondence, or connexion, though not an identity, with each other. Again, 
if we operate on the same octad © by the three coordinate characteristics of 
transposition w,, ,, w defined by the equations (21), we obtain these three new 
octads : 
OOS (Ap aas aA AS Aas 
@, 0, = (Ay Ag Age Ay Azy Age Ay Ai) 5 (110) 
,Q = (Ay Ag Alp Ago Ags Aas Ap Ay)’ 
to which correspond these three derived quaternions : 
